Learning Improvement by detection of incoherent experiments on Fuzzy Boolean Nets
JOSÉ ALBERTO TOMÉ
Instituto Superior Técnico/ INESC-ID
Rua Alves Redol nº9, 1000 Lisboa
PORTUGAL
JOÃO CARVALHO
Instituto Superior Técnico/ INESC-ID
Rua Alves Redol nº9, 1000 Lisboa
PORTUGAL
Keywords: Soft Computing , Learning , Fuzzy Boolean Neural Nets.
Abstract: - Fuzzy Boolean Nets are capable of learning by real experiments and they can implement a type of qualitative reasoning. There are certain contexts where incoherent experiments may arise with a certain probability, that is, where completely different consequents are originated from similar antecedent terms. This can spoil the teaching effectiveness and should be avoided. The problem is addressed and a solution is given using a relational supervising layer during the teaching phase.
1 Introduction[.]
Fuzzy Boolean Nets are capable of learning by real experiments and they can implement qualitative reasoning [8]. On such nets fuzziness is an emergent property in contrast with other known models where fuzziness is artificially introduced on neural nets or where neural components are inserted on the fuzzy systems [1, 3, 4, 5, 7]. The model is based on cards (or areas) of neurons, being these areas connected by meshes of weightless connections between antecedent neuron outputs and consequent neuron inputs. The individual connections are random, just like in nature, and neurons are binary, both in which concerns inputs and outputs. To each area a concept or variable is associated and the “value” of that concept or variable, when stimulated, is given by the activation ratio of that area (that is, the relation between activated -output “1”- neurons and the total number of neurons). Each consequent neuron basically samples each of the antecedent spaces using a limited number (say m) of the neuron’s outputs of that antecedent as its own inputs. This number, m, is much smaller then the number of neurons per area. This means that for rules with N-antecedent/1-Consequent each neuron has N.m inputs. The operations carried out by each neuron are performed with the classic Boolean operations (AND, OR), using as operands the inputs coming from antecedents and the Boolean internal state variables. These internal variables are established during a teaching phase and they may vary from 2N to (m+1)N bits, depending on how the m+1 possible counts of the m samples from each input are partitioned. It has been proved [8] that, from these micro operations (using local and limited information from individual neurons), it emerges a global or macro qualitative reasoning capability on the concepts, which can be expressedon the form of rules of type:
IF Antecedent1 is A1 AND Antecedent2 is A2 AND ….THEN Consequent is Ci,
In the expression, Antecedent1, Antecedent2,.., Consequent are variables or concepts and A1, A2; …, Ci are linguistic terms (or fuzzy sets) of these variables (such as, “small”, “high”, etc.). Notice that the number of linguistic terms equals the number of count sets from the above mentioned partition of m.
The proof [8] is achieved through the interpretation made to the relationship between the expressions of the antecedent and consequent activation ratios. The set of the consequent neurons is therefore named as reasoning layer.
During the learning phase the network is activated by a collection of experiments that will set or reset the individual neuron’s binary memories. For each experiment, a different input configuration (defined by the input areas specific samples) is presented to each and every of the consequent neurons. This configuration addresses one and only one the internal binary memories of each individual neuron. Updating of each binary memory value depends on its selection (or not) and on the logic value of the consequent neuron.
For the updating, one is considering the interesting case when non-selected binary memories maintain their state and selected binary memories take the value of consequent neuron, which corresponds to a kind of Grossberg based learning. It corresponds to the following updating equation, where p(t) is the probability of a given (corresponding to a given rule) internal memory being activated before the experiment, p(t+1) the same probability after the experiment, pout the probability of the consequent neuron to be activated and Pa the probability of addressing, in the same experiment, that internal memory:
p(t+1)-p(t) = Pa. (pout - p(t)) .This may be considered a Hebbian type of learning 2], if pre synaptic activity is associated with Pa, post synaptic activity is associated with pout and their correlation is related to p.
It has been proved [10] that the Net is capable of learning a set of different rules without cross-influence between different rules, and that the number of distinct rules that the system can effectively distinguish (in terms of different consequent terms) increases with the square root of the number m
It has also been proved that this model is a Universal Approximator [9], since it theoretically implements a Parzen Window estimator [6]. This means that these networks are capable to implement (approximate) any possible multi-input single-output function of the type: 0,1n0,1.
These results give the theoretical background to establish the capability of these simple binary networks to perform reasoning and effective learning based on real experiments. Since the nets present topologic similarities with natural systems and present also some of their important properties, it may be hypothesized that it may constitute a simple model for natural reasoning and learning. Also, as the emergent reasoning may be interpreted as fuzzy reasoning, it may also be hypothesized that natural reasoning is intrinsically fuzzy
2 Improving the Reasoning Layer
There are certain contexts where incoherent experiments may arise with a certain probability, that is, where completely different consequents are originated from similar antecedent terms. This can spoil the teaching effectiveness and should be avoided, in situations where an initial, more trustable phase of learning has taken place, but one wants to carry on with teaching/using experiments. Or, simply, the system wishes to be alerted when such contradictory experiments do occur.
It is the objective of this paper to present a natural (embedded on the network) solution to this problem. In order to do that the reasoning layer neurons are modified on a manner very similar to that used on the work related with the emotional layers on these networks [13].
The values taken by the linguistic terms on a given experiment are expressed by the binary memories activation ratios and these determine the consequent area activation ratio (which represents the consequent deffuzification). Then, any initial activation ratios (or corresponding to antecedent rule areas not taught enough) have no meaning but wrongly affect the consequent. In order to deal with this situation it is necessary to memorize, for each joint antecedent area of each neuron (corresponding to a given rule antecedent), not only the Boolean value of the that output but also extra binary information specifying if any thing at all has or not been taught - “0” if that joint antecedent has not been addressed or “1” if it has. In hardware terms this would mean two bits of memory per neuron and per joint antecedent area, instead of one. Let us designate these two bits by value (vl) and credit, (cr) respectively.
With this, the learnt knowledge of each rule, after the teaching phase, should now be interpreted not only by the active consequent activation ratio of that rule/joint antecedent area (defined as the ratio between the number of “ones” among every value bit of that rule on every consequent neuron and the number of “ones” among the respective credit bits), but also by the teaching ratio of the same rule (defined as the ratio of “ones” among the credit bits of that rule on every consequent neuron and the total number of consequent neurons, since each consequent neuron has one and only one credit bit Thus, after the teaching phase one knows which rules have been taught or not, just by evaluating the
teaching ratio of those rules.
Fig.1 A Neuron of the reasoning layer
per rule).
In figure 1, one neuron, of a network able to learn up to 9 different rules, is shown. In order to facilitate the explanation it is assumed that the counts from the 2 samples from each input have been partitioned into 2 sets (e.g. set of {0,1} and of {2} active inputs) and therefore limiting to 4 the number of possible rules. The figure shows a possible learning situation where the output binary value correspondent to that consequent neuron is “0”, and rule R0 has been addressed at that neuron – expressed by the black circle D0. Notice that, for a given experiment, one and only one of the rules can be addressed on each neuron, depending on the particular sampled input spaces. In addition it is supposed that the network, and this neuron in particular, has already learnt from past experiences. This learnt knowledge is expressed in the form of two bits for each rule: the credit bit, cr, which states if that rule has or not been taught to this neuron, respectively with value “1” (shadowed) or “0” (blank); and the value bit, vl that memorizes the binary value learnt if the respective cr bit is at “1”. Recall [8] that this learnt binary value is the consequent (output) bit value on the last teaching experiment where the same rule has been addressed. In the figure, only Rule 1 and Rule 3 have been learnt for that neuron, with respectively values “0” and “1”.
3 Detecting Incoherent Experiments
In order to detect incoherent experiments during extra learning, one may use similar Boolean networks to those used for learning and reasoning, as the one made of neurons of the type on the figure 1. These networks can be configured to perform relational operations [12], and in the actual context they use as input areas (variables) not the antecedent areas directly, rather they use areas formed with the memory bits of each neuron. Specifically, the areas: AVCi constituted by every neuron bit AND(Vi,Ci), for each rule i; area ADOi by bits AND(Di,OUTPUT) of every neuron, area Di by the bits Di and area CRi. By the bits Cri. The activation ratio of the first area represents the degree of membership of the assertion: “rule i has been highly taught with “1”s” [8], the second represents “in this experiment rule i is being addressed highly and its learnt consequent is high” and similarly for the other areas.
Each relational layer has the function of comparing what has already been taught for antecedent rule i with what the actual experiment wants to teach for the same rule, in case rule i has already learnt enough and is also being highly addressed on the experiment. If the net has already learnt with high credibility a given rule and if on a further experiment, which should be used just for “tuning” the learnt knowledge, the same rule antecedent is presented with a very different consequent, then the result of that experiment should be discarded or some type of action/alarm should be taken.
The relational operation concerned should then be of the following type:
IF((CRi is High AND Di is High) AND (AVCiis VERY DIFFERENT FROM ADOi ))
THEN DO NOT USE PRESENT EXPERIMENT FOR TEACHING
This rule can be read as follows: if rule i has been taught enough and is also being addressed enough on the teaching experiment and if the consequent values are too different do not use the experiment.
This kind of rule is easily implemented with Relational Fuzzy Boolean Nets [12], which use the bits Cri, Di, AVCi and ADOi to form the four input areas. As it is shown in [8], the linguistic terms above are obtained, implicitly, from the parameters used on the net, such as the number of inputs per neuron and per area.
Thus, a supervision layer composed of these relational layers (one for each rule), may be built in order to deal with such contradictory teaching actions.
Figure 2 tries to show the connections between the memory bits of the reasoning layer neurons and the supervision layer. Each neuron of the supervision layer samples the four input spaces shown (Cr, AVC, ADO and D); that is, each neuron of this layer receives input connections from a very limited number of the bits on those areas, exactly the same way the reasoning layer neurons do with the input areas. For instance, if this supervision layer uses m=4, each supervision layer neuron takes as inputs 4 randomly chosen Cr bits from the consequent neurons, and similarly 4 AVC bits, 4 ADO bits and 4 D bits.
Fig 2. The Supervision Layer
4 Conclusions
A class of Boolean nets with emerging fuzzy reasoning is capable of learning from experiments. There are certain circumstances, however, when following a more trustable teaching phase, possibly supervised, it is sought that the system continue to learn on a less trustable environment. This may happen because the initial experiments are considered insufficient to cover the entire antecedent domain or because the system is known to be somewhat time variant.
This paper presents an embedded network solution for the problem of avoiding teaching experiments that strongly disagree with previous more trustable knowledge.
A supervision layer, which input areas are formed with the internal memorized information of the basic reasoning layer neurons, is used. It is capable of detecting incongruent experiments that sharply differ from the previous learnt knowledge. This may be used to cancel teaching or to perform any other adequate action.
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[.]This work is partially supported by FCT project POSI/SRI/47188/2002